A store owner decides to put a coupon in the Sunday newspaper. The advertising firm she works believes that at least 20% but no more than 30% of people who shop at his store will use the coupon. The paper circulates 50,000 copies of the Sunday paper and ALL her customers get the Sunday paper. She believes that everyone shopping at her store receives the Sunday paper. What is the probability that 50% or more of the customers in a single day will have a coupon?

Question

A store owner decides to put a coupon in the Sunday newspaper. The advertising firm she works believes that at least 20% but no more than 30% of people who shop at his store will use the coupon. The paper circulates 50,000 copies of the Sunday paper and ALL her customers get the Sunday paper. She believes that everyone shopping at her store receives the Sunday paper. What is the probability that 50% or more of the customers in a single day will have a coupon?

Answer 1

To find the probability that 50% or more of the customers in a single day will have a coupon, we need to calculate the cumulative probability from 50% to 100%. We will use the binomial probability formula to solve this problem.

The binomial probability formula is:

P(x) = (nCx) * p^x * q^(n-x)

Where:
P(x) is the probability of getting exactly x successes.
n is the total number of trials.
x is the number of successful trials.
p is the probability of success for a single trial.
q is the probability of failure for a single trial.

In this case, the success is defined as using the coupon, and failure is defined as not using the coupon.

Let's break down the problem:

1. Total customers: Since all of the store owner's customers receive the Sunday paper, the total number of customers is equal to the circulation of the paper, which is 50,000.

2. Probability of using the coupon: The advertising firm believes that at least 20% but no more than 30% of people who shop at the store will use the coupon. We can assume a probability of 0.25 (25%) for this calculation.

3. Probability of not using the coupon: The complement of using the coupon is not using the coupon, which is 1 - 0.25 = 0.75 (75%).

Now, let's calculate the probability using the cumulative binomial probability:

P(x ≥ 50%) = P(x = 50%) + P(x = 51%) + ... + P(x = 100%)

P(x) = (nCx) * p^x * q^(n-x)

P(x ≥ 50%) = P(x = 50%) + P(x = 51%) + ... + P(x = 100%)

P(x = 50%) = (nCk) * p^k * q^(n-k) = (50000C25000) * (0.25^25000) * (0.75^(50000-25000))

Similarly, calculate P(x = 51%), P(x = 52%), ..., P(x = 100%) using the same formula.

Lastly, add up all these probabilities to get the cumulative probability:

P(x ≥ 50%) = P(x = 50%) + P(x = 51%) + ... + P(x = 100%)

By calculating these individual probabilities and summing them up, you will get the probability that 50% or more of the customers in a single day will have a coupon.